Classifications of commutative Banach algebras and Banach modules and its applications

分类的基础是设置几个条件，并考虑满足条件的类。本研究一直致力于阐明交换Banach代数和Banach模的本质：首先，根据给定的自然条件对它们进行分类，然后研究具体的代数和模是否属于分类群，以及特定的分类代数和模具有哪些不变性质。在此之前，基于上述思想，我们引入并研究了BSE-代数和BSE-Banach模各自的群。本文以局部紧交换群上的具体代数为研究对象，证明了紧交换群Γ的有限正则Borel测度集在加法下是闭的当且仅当Γ是离散的。如果G也是非离散局部紧交换群，则存在有限Reg…L(G)上具有自然谱的乘子算子不可分解的更多正则Borel测度。其次，我们证明了两个有单位的交换Banach代数之间的某些环同态是自动线性的。其中，我们证明了圆盘代数上的环上的同态是以包括素数理想的形式给出的。此外，Pfaffenberger和Phillips考虑了经典交换Gelfand定理的一个实数和单位情形，得到了两个表示定理。一种是将单位实交换Banach代数A表示为单位同态空间Φ_A上的连续函数的代数，另一种是将A表示为极大理想空间MA上的连续截面的代数。本文指出，对于非单位情形，类似的定理也成立，并证明了两个表示定理本质上相同。我们还证明了从实交换Banach代数到严格实交换Banach代数的环同态是由极大理想空间之间的某些连续映射来表示的。作为应用，我们考虑了以下问题，得到了许多有价值的结果：线性微分算子的Hyers-Ulam稳定性问题，椭圆型微分方程u“=f(t，u)的第三类边值问题，以及拓扑向量空间上闭凸集族与不等式闭凸集族的对应问题。

Classifications are based on setting several conditions and considering the classes to satisfy the conditions. This research has been focusing on clarifying the essence of commutative Banach algebras and Banach modules by the following idea : First, they would be classified according to the natural conditions settled, and then whether concrete algebras and modules belong to the classified groups or not, and what invariant properties the specific classified algebra and module have, might be investigated. Before this investigation, based on the above idea we have introduced and investigated the groups respective to BSE-algebras and BSE-Banach modules. In this research, paying attention to concrete algebras on locally compact abelian groups, we showed that the set of finite regular Borel measures with natural spectra for a compact abelian group Γ is closed under addition if and only if Γ is discrete. If also G is a non-discrete locally compact abelian group, then there exists a finite reg … More ular Borel measure with natural spectrum of which corresponding multiplier operator on L'(G) is not decomposable. We next proved that certain ring homomorphisms between two commutative Banach algebras with units are automatically linear. Among other things, we proved that ring homomor-phisms on the disk algebra into itself is given in terms including of prime ideals. Moreover Pfaffenberger and Phillips consider a real and unital case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real commutative Banach algebra A as an algebra of continuous functions on the unital homo-morphism space Φ_A. The other is to represent A as an algbera of continuous sections on the maximal ideal space M_A. In this note, we point out that similar theorems for non-unital case hold and show that two representation theorems are essentially identical. We also proved that ring homomorsphisms from real commutative Banach algebras into strictly real commutative Banach algebras are represented by certatain continuous mappings between the maximal ideal spaces.As an application, we considered the following problem, and obtained many valuable results : the Hyers-Ulam stability problem of linear differential operators, a third kind boundary value problem of elliptic differential equation u"= f(t,u) and the correspondence problem between the family of closed convex sets and that of inequalities on a topological vector space. Less